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R. Ge and Z. Xia, “A Type of Modified BFGS Algorithm with Rank Defects and Its Global Convergence in Convex Minimization,” Journal of Pure and Applied Mathematics: Advances and Applications, Vol. 3, No. 1, 2010, pp. 17-35.

has been cited by the following article:

• TITLE: Neural Network Approach for Solving Singular Convex Optimization with Bounded Variables

  AUTHORS: Rendong Ge, Lijun Liu, Yi Xu

  KEYWORDS: Neural Networks;Singular Nonlinear Optimization;Stationary Point;Augmented Lagrangian Function;Convergence; LaSalle’s Invariance Principle Plain

  JOURNAL NAME: Open Journal of Applied Sciences, Vol.3 No.3, July 17, 2013

  ABSTRACT: Although frequently encountered in many practical applications, singular nonlinear optimization has been always recognized as a difficult problem. In the last decades, classical numerical techniques have been proposed to deal with the singular problem. However, the issue of numerical instability and high computational complexity has not found a satisfactory solution so far. In this paper, we consider the singular optimization problem with bounded variables constraint rather than the common unconstraint model. A novel neural network model was proposed for solving the problem of singular convex optimization with bounded variables. Under the assumption of rank one defect, the original difficult problem is transformed into nonsingular constrained optimization problem by enforcing a tensor term. By using the augmented Lagrangian method and the projection technique, it is proven that the proposed continuous model is convergent to the solution of the singular optimization problem. Numerical simulation further confirmed the effectiveness of the proposed neural network approach.